(Solved): When a conduction current exists in the wires, a changing clectric field \( \vec{E} \) exists betw ...

When a conduction current exists in the wires, a changing clectric field \( \vec{E} \) exists between the plates of the capacitac. SOLUTION Conceptualize Figure (a) represents the circuit diagram for this situation. Figure (b) shows a close-up of the capacitor and the electric field between the plates. In fiqure (b), in which direction is the current flowing? Categorize We determine results using equations discussed in this section, so we categorize this example as problem. Analyze (Use the following as necessary: \( \omega_{r} \Delta V_{\max }, C_{1} \) and \( t_{1} \) ) Evaluate the angular frequency (in rad/s) of the source from \( \omega=2 \pi f \); \[ \omega=2 \pi f=\quad \mathrm{rad} / \mathrm{s} \] Use \( \Delta v_{C}=\Delta V_{\max } \sin (\omega t) \) to express the potential difference in volts across the capacitor as a function of time in seconds (Enter the angular frequency in rad/s): \[ \Delta v_{C}=\Delta V_{\max } \sin (\omega t)=30.0 \sin (\quad t) \] Use \( I_{d}=\frac{d q}{d t} \) to find the displacement current in amperes as a function of time. Note that the charge on the capacitor is \( q=C \Delta v_{c} \) : \[ i_{d}=\frac{d q}{d t}=\frac{d}{d t}\left(C \Delta v_{C}\right)=C \frac{d}{d t}\left(\Delta V_{\max } \sin (\omega t)\right) \] \( = \) Substitute numerical values to find \( i_{d} \) (in A) in terms of \( t \) (Do not include units in your answer.):